Matrices with the sign pattern of the Kronecker sum

The Kronecker sum is defined as $$A \oplus B = A \otimes I_m + I_n \otimes B,$$ where $\otimes$ is the Kronecker product, and $A$ is $n\times n$ and $B$ is $m \times m$. It has lots of nice properties, which stem from the fact that $\exp(A \oplus B) = \exp(A) \otimes \exp(B)$, where $\exp$ is the matrix exponential.

Matrices generated from a Kronecker sum have a particular pattern of zero and non-zero entries. I am interested in matrices that have this same pattern, but which cannot be expressed as a Kronecker sum. I want to know whether having this particular set of zero entries puts any identifiable constraints on the eigendecomposition and other linear algebra properties of the matrix.

One reason to think that there might be useful results here is that block matrices can be seen as matrices that have the same pattern of zero and non-zero entries as matrices generated from the Kronecker product. Since the Kronecker sum and Kronecker product are so closely related, and since there are so many useful results regarding block matrices, it seems reasonable to think there would be corresponding results for the Kronecker sum.

In particular, I'm interested in the case where all of the non-zero elements are positive, except for possibly the diagonal elements. This sign pattern arises from the logarithm of the Kronecker product of matrices with non-negative entries.